On the topology of the magnetic lines of large solutions to the Magnetohydrodynamic equations in $\mathbb{R}^3$

TL;DR

This paper introduces a new class of large solutions to the magnetohydrodynamic equations in three-dimensional space, demonstrating magnetic reconnection through topological changes in magnetic field lines, based on counting hyperbolic critical points.

Contribution

It establishes the existence of large global solutions with a specific smallness condition on their initial data difference, and proves magnetic reconnection with topological changes in magnetic lines.

Findings

Existence of large global solutions with controlled initial data difference.

Magnetic reconnection demonstrated via topological changes in magnetic lines.

Counting hyperbolic critical points provides a stable method for analyzing topology.

Abstract

The purpose of this article is twofold: first, we introduce a new class of global strong solutions to the magnetohydrodynamic system in $\mathbb{R}^3$ with initial data $(u_0,b_0)$ of arbitrarily large size in any critical space. To do so, we impose a smallness condition on the difference $u_0-b_0$. Then we use this result to prove magnetic reconnection for a suitable class of (large) solutions. With this, we mean a change of topology of the integral lines of the magnetic field $b$ under the evolution. The proof relies on counting the number of hyperbolic critical points of the solutions, and this instance is structurally stable.